Re: Linear--structure or property?

["Fred E.J. Linton" <fejlinton@usa.net>]

Three items with which to follow up on my earlier note,

> For instance, take the additive group of 2x2 matrices with
> integer entries (or entries from any semiring) and notice that,
> apart from the usual matrix multiplication, there is also
> the sophomoric, or pointwise, multiplication ...:

1. In one of the poster sessions at the recent Madrid ICM2006,
Camarero, Etayo, Rovira, and Santamaria remind us that the ordinary
real plane R^2 (with usual vector addition) admits at least

> three distinguished real algebras ... as follows: the set
> {a+bi: a, b {/element} R} with i^2 = -1, +1, 0, i.e., the
> complex, double, and dual numbers.

(ICM2006 Abstracts, p. 42)

2. Yefim Katsov (in a telephone conversation) has pointed out
that in any reasonable lattice-ordered (semi-)group, where
a + (b ^ c) = (a+b) ^ (a+c) and a + (b v c) = (a+b) v (a+c),
one gets two different (semi-)ring structures by using: 
as product, the (semi-)group composition + ; and 
as sum, in one case the lattice meet ^ , alternatively, the join v .

3. A many-objects version can be concocted from example 2 above
by stirring it up with a variant form of Lawvere's observations
about metric spaces being categories enriched over an appropriately
structured closed monoidal version of the poset R+ of nonnegative
real numbers ( order relation > , tensor product + , unit 0 ,
internal hom the positive part of b-a ). 

In detail, if X is a metric space with metric d, consider the
ordinary category _X_ whose objects are the points of X while its
homsets _X_(p, q) are given by the principal filter (in (R, </= ) )
generated by d(p, q):

_X_(p, q) = { x {/element} R: x >/= d(p, q) } .

Composition _X_(p, q) x _X_(q, r) can clearly be given by
sending (x, y) to x+y : for identity map is always the number 0,
and whenever x >/= d(p, q) and y >/= d(q, r) we must also have
x+y >/= d(p, q) + d(q, r) >/= d(p, r) .

Thus, arithmetic addition provides a composition rule for _X_ ,
and both real sup and real inf can serve as commutative semigroup
structures (across which composition distributes) on the homsets.

Of course no one will claim _X_ has any finite (co)products; but,
anyway, here any enrichment of _X_ over semigroups is clearly an
added item of structure, and not a property of _X_ .

-- Fred (and pardon, please, the crude ASCII/TeX symbology)

PS: Katsov has also pointed out that a marvelous little New Yorker
piece of Fields Medal gossip, turning around Yau, Perelman, Hamilton,
and the Poincare conjecture, can be found on the web (for those who
don't take the New Yorker, or even those who do) at:

http://www.newyorker.com/printables/fact/060828fa_fact2 .

-- F.